Sophie, I think that the inequalities from this Lecture 27 are the "correct" ones. A nice way to remember which way the unit condition should go is to note that there the greater or equal side is of the form \\(f(-)\\), both in \\(f(x) \otimes f(x') \leq f(x\otimes x')\\) and in \\(I \leq f(I)\\). These two inequalities are like the multiplication and the unit of a monoid: in the case of the multiplication, you have *two* things going in and *one* thing coming out, which corresponds to two \\(f\\)'s on the left and one on the right. The unit of a monoid is just a constant, which means that it has *no* thing going in and again *one* thing coming out; just like the \\(f\\)'s in \\(I\leq f(I)\\).
In case that all this sounds a bit mysterious or confusing to you now, let me just say that it will become clear further along the course! (Assuming that lax monoidal functors will be covered.)